Odd-coordinate kernel conjecture for singular io-decomposable Riordan graphs

Let $G_n(g,zg)$ be a singular io-decomposable Riordan graph, let ${\cal A}_n$ be its adjacency matrix, and let $X=[x_1,x_2,\ldots,x_n]^T$ be an eigenvector with eigenvalue $0$. Define the odd-index and even-index subvectors by

$$X_o=[x_1,x_3,\ldots,x_{2\lceil n/2\rceil-1}]^T,$$ $$X_e=[x_2,x_4,\ldots,x_{2\lfloor n/2\rfloor}]^T.$$

Odd-coordinate kernel conjecture. Then

$$X_o=O_{\lceil n/2\rceil}.$$

This claim is motivated by Sage computations and, together with the preceding block-matrix lemma, would constrain the zero-eigenspace of every singular io-decomposable Riordan graph. No proof or resolution is supplied.